com www.GOALias.blogspot.com

www.GOALias.blogspot.comvalue of Q, the smaller is the value of 2Δω or the bandwidth and sharper2is the resonance. Using ω 0 = 1/LC, Eq. [7.36(c)] can be equivalentlyexpressed as Q = 1/ω 0CR.We see from Fig. 7.15, that if the resonance is less sharp, not only isthe maximum current less, the circuit is close to resonance for a largerrange Δω of frequencies and the tuning of the circuit will not be good. So,less sharp the resonance, less is the selectivity of the circuit or vice versa.From Eq. (7.36), we see that if quality factor is large, i.e., R is low or L islarge, the circuit is more selective.Alternating CurrentExample 7.6 A resistor of 200 Ω and a capacitor of 15.0 μF areconnected in series to a 220 V, 50 Hz ac source. (a) Calculate thecurrent in the circuit; (b) Calculate the voltage (rms) across theresistor and the capacitor. Is the algebraic sum of these voltagesmore than the source voltage? If yes, resolve the paradox.SolutionGiven−6R = 200 Ω , C = 15.0 μ F = 15.0 × 10 FV = 220 V, ν = 50 Hz(a) In order to calculate the current, we need the impedance of thecircuit. It isZ = R + X = R + (2 πνC)2 2 2 −2C−= (200 Ω ) + (2× 3.14× 50×10 F)2 6 −22 2= (200 Ω ) + (212 Ω)= 291.5 ΩTherefore, the current in the circuit isV 220 VI = = = 0.755 AZ 291.5 Ω(b) Since the current is the same throughout the circuit, we haveVR= I R = (0.755 A)(200 Ω ) = 151 VVC= I XC= (0.755 A)(212.3 Ω ) = 160.3 VThe algebraic sum of the two voltages, V Rand V Cis 311.3 V which ismore than the source voltage of 220 V. How to resolve this paradox?As you have learnt in the text, the two voltages are not in the samephase. Therefore, they cannot be added like ordinary numbers. Thetwo voltages are out of phase by ninety degrees. Therefore, the totalof these voltages must be obtained using the Pythagorean theorem:2 2VR+ C= VR+VC= 220 VThus, if the phase difference between two voltages is properly takeninto account, the total voltage across the resistor and the capacitor isequal to the voltage of the source.EXAMPLE 7.6251